[PPT] - Calculational HoTT International Conference on Homotopy Type Theory PowerPoint Presentation

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Calculational HoTT International Conference on Homotopy Type Theory (HoTT 2019) Carnegie Mellon University August 12 to 17, 2019

Bernarda Aldana, Jaime Bohorquez, Ernesto Acosta Escuela Colombiana de Ingenier´ ıa Bogot´ a, Colombia

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Content

1 A few initial words 2 Brief description of CL 3 The problem 4 Deductive chains 5 Calculational HoTT 6 A deduction 7 Conclusions

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Presentation

What we do is to rewrite math topics using Calculational Logic (CL),

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Presentation

What we do is to rewrite math topics using Calculational Logic (CL), as there is a large community rewriting math in terms of HoTT.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Presentation

What we do is to rewrite math topics using Calculational Logic (CL), as there is a large community rewriting math in terms of HoTT. We ended up trying to interpret HoTT in terms of CL.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Presentation

What we do is to rewrite math topics using Calculational Logic (CL), as there is a large community rewriting math in terms of HoTT. We ended up trying to interpret HoTT in terms of CL. The result: “Calculational HoTT”(arXiv:1901.08883v2), a joint work with Bernarda Aldana and Jaime Bohorquez.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Equational axioms and Leibniz rules

Brief description of CL. Main feature: CL axioms are logical equations A ≡ B, C ≡ D, . . . CL is an equational logical system CL inference rules are Leibniz’s rules E[x/A] A ≡ B E[x/B] E[x/B] A ≡ B E[x/A]

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Calculations

Derivations in CL are deduction trees of the form: E1 A ≡ B E2 C ≡ D E3 E ≡ F E4 where A through F are subformulas of the corresponding Ei.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Calculations

Derivations in CL are deduction trees of the form: E1 A ≡ B E2 C ≡ D E3 E ≡ F E4 where A through F are subformulas of the corresponding Ei. This deduction tree, written vertically, is what Lifschitz called ‘Calculation’[Lifs]: E1 ⇔ A ≡ B E2 ⇔ C ≡ D E3 ⇔ E ≡ F E4 which derives E1 ≡ E4 Double arrows stand for the bidi- rectionality of Leibniz rules

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Calculations

Derivations in CL are deduction trees of the form: E1 A ≡ B E2 C ≡ D E3 E ≡ F E4 where A through F are subformulas of the corresponding Ei. This deduction tree, written vertically, is what Lifschitz called ‘Calculation’[Lifs]: E1 ⇔ A ≡ B E2 ⇔ C ≡ D E3 ⇔ E ≡ F E4 which derives E1 ≡ E4 Double arrows stand for the bidi- rectionality of Leibniz rules There are sound and complete calculational versions of both, classical (CCL) and intuitionistic (ICL) first order logic.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Embeddings

The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Embeddings

The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory We pose ourself the following question: Is it possible to embed ICL into HoTT?

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Embeddings

The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory We pose ourself the following question: Is it possible to embed ICL into HoTT? We concentrated in

establishing a linear calculation format as an instrument to understand

proofs in HoTT book, and

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Embeddings

The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory We pose ourself the following question: Is it possible to embed ICL into HoTT? We concentrated in

establishing a linear calculation format as an instrument to understand

proofs in HoTT book, and

identify and derive equational judgments in HoTT.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Embeddings

The problem Curry-Howard isomorphism embeds intuitionistic predicate logic into dependent type theory We pose ourself the following question: Is it possible to embed ICL into HoTT? We concentrated in

establishing a linear calculation format as an instrument to understand

proofs in HoTT book, and

identify and derive equational judgments in HoTT.

Note: We expected to be more comfortable with a linear calculation format as an instrument to understand proofs in HoTT book.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

First: Definition of deductive chains.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

First: Definition of deductive chains. A → B <: A ❀ B (read A leads to B) stands temporarily for one of the judgments A ≡ B

r A ≃ B <:

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

First: Definition of deductive chains. A → B <: A ❀ B (read A leads to B) stands temporarily for one of the judgments A ≡ B

r A ≃ B <:

It is easy to prove the following transitivity rule scheme A1 ❀ A2 A2 ❀ A3 A1 ❀ A3 where the conclusion corresponds to

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

First: Definition of deductive chains. A → B <: A ❀ B (read A leads to B) stands temporarily for one of the judgments A ≡ B

r A ≃ B <:

It is easy to prove the following transitivity rule scheme A1 ❀ A2 A2 ❀ A3 A1 ❀ A3 where the conclusion corresponds to A1 → A3 <: if at least one of the premises is a judgment of the form A → B <:

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

First: Definition of deductive chains. A → B <: A ❀ B (read A leads to B) stands temporarily for one of the judgments A ≡ B

r A ≃ B <:

It is easy to prove the following transitivity rule scheme A1 ❀ A2 A2 ❀ A3 A1 ❀ A3 where the conclusion corresponds to A1 → A3 <: if at least one of the premises is a judgment of the form A → B <: A1 ≃ A3 <: if none of the premises is of the form A → B <: and at least one is of the form A ≃ B <:

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

First: Definition of deductive chains. A → B <: A ❀ B (read A leads to B) stands temporarily for one of the judgments A ≡ B

r A ≃ B <:

It is easy to prove the following transitivity rule scheme A1 ❀ A2 A2 ❀ A3 A1 ❀ A3 where the conclusion corresponds to A1 → A3 <: if at least one of the premises is a judgment of the form A → B <: A1 ≃ A3 <: if none of the premises is of the form A → B <: and at least one is of the form A ≃ B <: A1 ≡ A3 if all the premises are of the form A ≡ B

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

By induction we have the following derivation . . . a : A1 . . . A1 ❀ A2 · · · . . . An−1 ❀ An An <: .

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

By induction we have the following derivation . . . a : A1 . . . A1 ❀ A2 · · · . . . An−1 ❀ An An <: . which may be represented vertically by the following format-scheme An ⇆ · · · An−1 . . . A2 ⇆ · · · A1

∧ :

· · · a which we called a deductive chain.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

The links in this format-scheme are B ⇆ A

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

The links in this format-scheme are B ⇆ A consequence link B ← : ; evidence A

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

The links in this format-scheme are B ⇆ A consequence link B ← : ; evidence A equivalence link B ≡ evidence A

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

The links in this format-scheme are B ⇆ A consequence link B ← : ; evidence A equivalence link B ≡ evidence A h-equivalence link B ≃ : ; evidence A

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Deductive chains

The links in this format-scheme are B ⇆ A consequence link B ← : ; evidence A equivalence link B ≡ evidence A h-equivalence link B ≃ : ; evidence A The link at the bottom of the deductive chain is called inhabitation link.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Quantified proposition notation

Unified notation for operationals (Qx:T | range · term)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Quantified proposition notation

Unified notation for operationals (Qx:T | range · term) Examples:

Summation:

(Σi:N | 1 ≤ i ≤ 3 · i2) = 12 + 22 + 32

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Quantified proposition notation

Unified notation for operationals (Qx:T | range · term) Examples:

Summation:

(Σi:N | 1 ≤ i ≤ 3 · i2) = 12 + 22 + 32

Logical operationals (universal and existential quantifiers)

(∀x:T | range · term) for conjunction, (∃x:T | range · term) for disjunction.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Quantified proposition notation

Unified notation for operationals (Qx:T | range · term) Examples:

Summation:

(Σi:N | 1 ≤ i ≤ 3 · i2) = 12 + 22 + 32

Logical operationals (universal and existential quantifiers)

(∀x:T | range · term) for conjunction, (∃x:T | range · term) for disjunction. [Trade] rules (∀x:T | P · Q) ≡ (∀x:T · P ⇒Q) (∃x:T | P · Q) ≡ (∃x:T · P ∧Q)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

Second: identify and derive equational judgments of HoTT corresponding to axioms and theorems of ICL:

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

Second: identify and derive equational judgments of HoTT corresponding to axioms and theorems of ICL: [One-Point]: (∀x:T | x=a · P) ≡ P[a/x] (∃x:T | x=a · P) ≡ P[a/x] (ICL)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

Second: identify and derive equational judgments of HoTT corresponding to axioms and theorems of ICL: [One-Point]: (∀x:T | x=a · P) ≡ P[a/x] (∃x:T | x=a · P) ≡ P[a/x] (ICL)

x:A
p:x=a P(x, p) ≃ P(a, refla)<:
x:A
p:x=a P(x, p) ≃ P(a, refla)<:

(HoTT)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

Second: identify and derive equational judgments of HoTT corresponding to axioms and theorems of ICL: [One-Point]: (∀x:T | x=a · P) ≡ P[a/x] (∃x:T | x=a · P) ≡ P[a/x] (ICL)

x:A
p:x=a P(x, p) ≃ P(a, refla)<:
x:A
p:x=a P(x, p) ≃ P(a, refla)<:

(HoTT) [Equality]: (∀x, y:T | x=y · P) ≡ (∀x:T · P[x/y]) (∃x, y:T | x=y · P) ≡ (∃x:T · P[x/y]) (ICL)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

Second: identify and derive equational judgments of HoTT corresponding to axioms and theorems of ICL: [One-Point]: (∀x:T | x=a · P) ≡ P[a/x] (∃x:T | x=a · P) ≡ P[a/x] (ICL)

x:A
p:x=a P(x, p) ≃ P(a, refla)<:
x:A
p:x=a P(x, p) ≃ P(a, refla)<:

(HoTT) [Equality]: (∀x, y:T | x=y · P) ≡ (∀x:T · P[x/y]) (∃x, y:T | x=y · P) ≡ (∃x:T · P[x/y]) (ICL)

x,y:A
p:x=y P(x, y, p) ≃

x:A P(x, x, reflx)<:

x,y:A
p:x=y P(x, y, p) ≃

x:A P(x, x, reflx)<:

(HoTT)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

[Range Split]: (∀x:T | P ∨ Q · R) ≡ (∀x:T | P · R) ∧ (∀x:T | Q · R) (∃x:T | P ∨ Q · R) ≡ (∃x:T | P · R) ∨ (∃x:T | Q · R)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

[Range Split]: (∀x:T | P ∨ Q · R) ≡ (∀x:T | P · R) ∧ (∀x:T | Q · R) (∃x:T | P ∨ Q · R) ≡ (∃x:T | P · R) ∨ (∃x:T | Q · R)

x:A+B P(x) ≃

x:A P(inl(x)) × x:B P(inr(x))<:

x:A+B P(x) ≃

x:A P(inl(x)) + x:B P(inr(x))<:

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

[Range Split]: (∀x:T | P ∨ Q · R) ≡ (∀x:T | P · R) ∧ (∀x:T | Q · R) (∃x:T | P ∨ Q · R) ≡ (∃x:T | P · R) ∨ (∃x:T | Q · R)

x:A+B P(x) ≃

x:A P(inl(x)) × x:B P(inr(x))<:

x:A+B P(x) ≃

x:A P(inl(x)) + x:B P(inr(x))<:

[Term Split]: (∀x:T | P · Q ∧ R) ≡ (∀x:T | P · Q) ∧ (∀x:T | P · R) (∃x:T | P · Q ∨ R) ≡ (∃x:T | P · Q) ∨ (∃x:T | P · R)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quantified axioms and theorems

[Range Split]: (∀x:T | P ∨ Q · R) ≡ (∀x:T | P · R) ∧ (∀x:T | Q · R) (∃x:T | P ∨ Q · R) ≡ (∃x:T | P · R) ∨ (∃x:T | Q · R)

x:A+B P(x) ≃

x:A P(inl(x)) × x:B P(inr(x))<:

x:A+B P(x) ≃

x:A P(inl(x)) + x:B P(inr(x))<:

[Term Split]: (∀x:T | P · Q ∧ R) ≡ (∀x:T | P · Q) ∧ (∀x:T | P · R) (∃x:T | P · Q ∨ R) ≡ (∃x:T | P · Q) ∨ (∃x:T | P · R)

x:A(P(x) × Q(x)) ≃

x:A P(x) × x:A Q(x)<:

x:A(P(x) + Q(x)) ≃

x:A P(x) + x:A Q(x)<:

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quatified axioms and theorems

[Translation] (∀x:J | P · Q) ≡ (∀y:K | P[f(y)/x] · Q[f(y)/x]) (∃x:J | P · Q) ≡ (∃y:K | P[f(y)/x] · Q[f(y)/x]), where f is a bijection that maps values of type K to values of type J.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quatified axioms and theorems

[Translation] (∀x:J | P · Q) ≡ (∀y:K | P[f(y)/x] · Q[f(y)/x]) (∃x:J | P · Q) ≡ (∃y:K | P[f(y)/x] · Q[f(y)/x]), where f is a bijection that maps values of type K to values of type J. [Congruence] (∀x:T | P · Q ≡ R) ⇒ ((∀x:T | P · Q) ≡ (∀x:T | P · R)) (∀x:T | P · Q ≡ R) ⇒ ((∃x:T | P · Q) ≡ (∃x:T | P · R))

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quatified axioms and theorems

[Translation] (∀x:J | P · Q) ≡ (∀y:K | P[f(y)/x] · Q[f(y)/x]) (∃x:J | P · Q) ≡ (∃y:K | P[f(y)/x] · Q[f(y)/x]), where f is a bijection that maps values of type K to values of type J. [Congruence] (∀x:T | P · Q ≡ R) ⇒ ((∀x:T | P · Q) ≡ (∀x:T | P · R)) (∀x:T | P · Q ≡ R) ⇒ ((∃x:T | P · Q) ≡ (∃x:T | P · R)) [Antecedent] R ⇒ (∀x:T | P · Q) ≡ (∀x:T | P · R ⇒ Q) R ⇒ (∃x:T | P · Q) ≡ (∃x:T | P · R ⇒ Q) when there are not free occurrences of x in R.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

ICL quatified axioms and theorems

[Translation] (∀x:J | P · Q) ≡ (∀y:K | P[f(y)/x] · Q[f(y)/x]) (∃x:J | P · Q) ≡ (∃y:K | P[f(y)/x] · Q[f(y)/x]), where f is a bijection that maps values of type K to values of type J. [Congruence] (∀x:T | P · Q ≡ R) ⇒ ((∀x:T | P · Q) ≡ (∀x:T | P · R)) (∀x:T | P · Q ≡ R) ⇒ ((∃x:T | P · Q) ≡ (∃x:T | P · R)) [Antecedent] R ⇒ (∀x:T | P · Q) ≡ (∀x:T | P · R ⇒ Q) R ⇒ (∃x:T | P · Q) ≡ (∃x:T | P · R ⇒ Q) when there are not free occurrences of x in R. [Leibniz principles] (∀x, y:T | x = y · f(x) = f(y)) (∃x, y:T | x = y · P(x) ≡ P(y)) where f is a function that maps values of type T to values of any other type and P is a predicate.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Equational judgments in HoTT

[Translation]

x:A P(x) ≃

y:B P(g(y))<:

x:A P(x) ≃

y:B P(g(y))<:

where g is an inhabitant of B ≃ A.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Equational judgments in HoTT

[Translation]

x:A P(x) ≃

y:B P(g(y))<:

x:A P(x) ≃

y:B P(g(y))<:

where g is an inhabitant of B ≃ A. [Congruence]

x:A(P(x) ≃ Q(x)) → (

x:A P(x) ≃ x:A Q(x))<:

x:A(P(x) ≃ Q(x)) → (

x:A P(x) ≃ x:A Q(x))<:

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Equational judgments in HoTT

[Translation]

x:A P(x) ≃

y:B P(g(y))<:

x:A P(x) ≃

y:B P(g(y))<:

where g is an inhabitant of B ≃ A. [Congruence]

x:A(P(x) ≃ Q(x)) → (

x:A P(x) ≃ x:A Q(x))<:

x:A(P(x) ≃ Q(x)) → (

x:A P(x) ≃ x:A Q(x))<:

[Antecedent] (R →

x:A Q(x)) ≃ x:A(R → Q(x))<:

x:A(R → Q(x)) → (R →

x:A Q(x))<:

when R does not depend on x.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Equational judgments in HoTT

[Translation]

x:A P(x) ≃

y:B P(g(y))<:

x:A P(x) ≃

y:B P(g(y))<:

where g is an inhabitant of B ≃ A. [Congruence]

x:A(P(x) ≃ Q(x)) → (

x:A P(x) ≃ x:A Q(x))<:

x:A(P(x) ≃ Q(x)) → (

x:A P(x) ≃ x:A Q(x))<:

[Antecedent] (R →

x:A Q(x)) ≃ x:A(R → Q(x))<:

x:A(R → Q(x)) → (R →

x:A Q(x))<:

when R does not depend on x. [Leibniz principles]

x,y:A

x=y → f(x)=f(y)<:

x,y:A

x=y → P(x)≃P(y)<: where f :A → B and P :A → U is a type family.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

I will derive the judgment (

x:A
y:B(x)

P((x, y))) ≃

g:

x:A B(x)

P(g) <: (1) which corresponds to the homotopic equivalence version of the Σ induction

perator.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

I will derive the judgment (

x:A
y:B(x)

P((x, y))) ≃

g:

x:A B(x)

P(g) <: (1) which corresponds to the homotopic equivalence version of the Σ induction

perator.
Note. The ICL theorem corresponding to (1), when P is a non-dependent

type, is (∀x:T | B · P) ≡ (∃x:T · B) ⇒ P where x does not occur free in P. This motivate us to call the equivalence Σ-[Consequent] rule.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Recall that the Σ-induction operator σ : (

x:A
y:B(x)

P((x, y))) →

g:

x:A B(x)

P(g) is defined by σ(u)((x, y)) :≡ u(x)(y).

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Recall that the Σ-induction operator σ : (

x:A
y:B(x)

P((x, y))) →

g:

x:A B(x)

P(g) is defined by σ(u)((x, y)) :≡ u(x)(y). Let Φ : (

g:

x:A B(x)

P(g)) →

x:A
y:B(x)

P((x, y)) be defined by Φ(v)(x)(y) :≡ v((x, y)).

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Let u be an inhabitant of

x:A

y:B(x)

P((x, y)),

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Let u be an inhabitant of

x:A

y:B(x)

P((x, y)), then Φ(σ(u)) = u

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Let u be an inhabitant of

x:A

y:B(x)

P((x, y)), then Φ(σ(u)) = u ≃ : ;Function extensionality

x:A
y:B(x)

Φ(σ(u))(x)(y) = u(x)(y)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Let u be an inhabitant of

x:A

y:B(x)

P((x, y)), then Φ(σ(u)) = u ≃ : ;Function extensionality

x:A
y:B(x)

Φ(σ(u))(x)(y) = u(x)(y) ≡ Φ(σ(u)) ≡ σ(u)((x, y)) ≡ u(x)(y)

x:A
y:B(x)

u(x)(y) = u(x)(y)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Let u be an inhabitant of

x:A

y:B(x)

P((x, y)), then Φ(σ(u)) = u ≃ : ;Function extensionality

x:A
y:B(x)

Φ(σ(u))(x)(y) = u(x)(y) ≡ Φ(σ(u)) ≡ σ(u)((x, y)) ≡ u(x)(y)

x:A
y:B(x)

u(x)(y) = u(x)(y)

∧ :

hu(x)(y) :≡ reflu(x)(y) hu

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Let u be an inhabitant of

x:A

y:B(x)

P((x, y)), then Φ(σ(u)) = u ≃ : ;Function extensionality

x:A
y:B(x)

Φ(σ(u))(x)(y) = u(x)(y) ≡ Φ(σ(u)) ≡ σ(u)((x, y)) ≡ u(x)(y)

x:A
y:B(x)

u(x)(y) = u(x)(y)

∧ :

hu(x)(y) :≡ reflu(x)(y) hu Then Φ ◦ σ is homotopic to the identity function of

x:A

y:B(x)

P((x, y)).

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Conversely, let v be an inhabitant of

g:

x:A B(x) P(g),

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Conversely, let v be an inhabitant of

g:

x:A B(x) P(g), then

σ(Φ(v)) = v

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Conversely, let v be an inhabitant of

g:

x:A B(x) P(g), then

σ(Φ(v)) = v ≃ : ;Function extensionality

g:

x:A B(x)

σ(Φ(v))(g) = v(g)

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Conversely, let v be an inhabitant of

g:

x:A B(x) P(g), then

σ(Φ(v)) = v ≃ : ;Function extensionality

g:

x:A B(x)

σ(Φ(v))(g) = v(g) ← : σ′

x:A
y:B(x)

σ(Φ(v))(x, y) = v((x, y))

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Conversely, let v be an inhabitant of

g:

x:A B(x) P(g), then

σ(Φ(v)) = v ≃ : ;Function extensionality

g:

x:A B(x)

σ(Φ(v))(g) = v(g) ← : σ′

x:A
y:B(x)

σ(Φ(v))(x, y) = v((x, y)) ≡ σ(Φ(v))((x, y)) ≡ Φ(v)(x)(y) ≡ v((x, y))

x:A
y:B(x)

v((x, y)) = v((x, y))

SLIDE 65

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Conversely, let v be an inhabitant of

g:

x:A B(x) P(g), then

σ(Φ(v)) = v ≃ : ;Function extensionality

g:

x:A B(x)

σ(Φ(v))(g) = v(g) ← : σ′

x:A
y:B(x)

σ(Φ(v))(x, y) = v((x, y)) ≡ σ(Φ(v))((x, y)) ≡ Φ(v)(x)(y) ≡ v((x, y))

x:A
y:B(x)

v((x, y)) = v((x, y))

∧ :

hv(x, y) :≡ reflv(x,y) hv

SLIDE 66

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

A deduction

Conversely, let v be an inhabitant of

g:

x:A B(x) P(g), then

σ(Φ(v)) = v ≃ : ;Function extensionality

g:

x:A B(x)

σ(Φ(v))(g) = v(g) ← : σ′

x:A
y:B(x)

σ(Φ(v))(x, y) = v((x, y)) ≡ σ(Φ(v))((x, y)) ≡ Φ(v)(x)(y) ≡ v((x, y))

x:A
y:B(x)

v((x, y)) = v((x, y))

∧ :

hv(x, y) :≡ reflv(x,y) hv So, σ ◦ Φ is homotopic to the identity function of

g:

x:A B(x) P(g).

This proves the Σ-[Consequent] rule.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Example

Application of Π-translation rule (to prove isSet(N) <:). We can use the translation rule to prove isSet(N) <: In fact, let Φ : m = n → code(m, n) be defined by Φ :≡ encode(m, n) and let Ψ : code(m, n) → m = n be defined by Ψ :≡ decode(m, n). Then, isSet(N) ≡ Definition of isSet

m,n:N
p,q:m=n

p = q ≃ Π-translation rule ; m = n ≃ code(m, n)

m,n:N
s,t:code(m,n)

Ψ(s) = Ψ(t)

∧ :

See definition of h below h

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Conclusions

Conclusions:

SLIDE 69

A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Conclusions

Conclusions:

1 Deductive chains are really formal linear tools to prove theorems in

HoTT.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Conclusions

Conclusions:

1 Deductive chains are really formal linear tools to prove theorems in

HoTT.

2 There is an embedding of ICL in HOTT. In particular we found that

the Eindhoven quantifiers correspond to the main dependent types in HoTT.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

Conclusions

Conclusions:

1 Deductive chains are really formal linear tools to prove theorems in

HoTT.

2 There is an embedding of ICL in HOTT. In particular we found that

the Eindhoven quantifiers correspond to the main dependent types in HoTT.

3 We found strong evidence that it is possible to restate the whole of

HoTT giving equality and homotopic equivalence a preeminent role, both, axiomatically and proof-theoretically.

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A few initial words Brief description of CL The problem Deductive chains Calculational HoTT A deduction Conclusions

T. Univalent Foundations Program.